Basics of Bayesian Statistics

We can calcualte “What is the probability of event A given event B?”" using Bayesian. This is called conditional probablity

Eg: False positive or False negative rates.

Conditional Probability and Bayes’ rule

Polls : Have you even used dating websites?

Bayes rule application. Recalculating using bayes rule.

Bayes Rule and Diagnostic Testing

US Militray early HIV testing in military

  1. Elisa screen
  2. if positive, two more ELISA
  3. if either positive, two western blot assays
  4. if positive, HIV infection

ELISA -sensitivity (true positive): 93% p(+/HIV) = 0.93 -specificity (true negative) : 99% p(-/No HIV) = 0.99

Western blot -sensitivity (true positive): 99.9% -specificity (true negative) : 99.1%

prevalence : 1.48 / 1000 people p(HIV) = 0.0148 which is prior

Bayes-tree

Bayes-tree

This updating scheme we have here is general property of the Bayesian models

Bayesian and Frequentist definitions of probabbility

Frequentist: Its relative frequency in large number of trials Bayesian : Probability of an event happening is equated to another event

Confidence Interval : The proportion of random samples of size n from the same population that produce confidence interval that contain the true population parameter.

Credible Interval : We can express the true parameter not as a fixed value but with a probablity This will let us contruct something like a credible intervals expect we can make probabilistic statements about the paramenter falling within that range

Inference for proportions :

RU-486 effective or not? 40 women are divided into two groups with one taking RU 486 and the other standard therapy.4/20 got pregnant where RU 486 used and 16/20 got pregnent where stadard methods are used. How strongly does the data indicate RU 486 is effective? This is a two proportion problem but we can frame it as one proportion test as following: How likely that 4 pregnencies occur in treatment group? p = probability that a given pregnency comes from a treatment group

H\(_{0}\) : p = 0.5 - No difference, pregnency is equally likely to come from the treatment or control group

H\(_{A}\) : p > 0.5 - treatment is more effective, a pregnency is less likely to come from the treatment group

Bayesian Approach :

Doubts

What is likelihood in the Bayes tree? I think it is the sensitivity and specificity of the test.